Fatigue crack growth rate data from center-cracked panels using Newman's crack closure model, from compact specimens using Eason 's R-ratio expression, and from bend specimens using the model discussed in this paper are all shown to fall along the same straight line (on log-log paper) when plotted versus ΔK eff, even though crack closure. The fatigue crack growth rates for Al and Ti are much more rapidthan steel for a given?K. However, when normalized by Young’s Modulus all metals exhibit about the same behavior. Crack growth rates of metals ∆K E. Crack-growth results, based on studies on “long” (3mm) cracks, show fatigue-crack propagation rates to be markedly power-law dependent on the applied stress-intensity range, ΔK, with a.

1. Fatigue Crack Growth Testing
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Paris' law (also known as the Paris-Erdogan equation) is a crack growth equation that gives the rate of growth of a fatigue crack. The stress intensity factor${\displaystyle K}$ characterises the load around a crack tip and the rate of crack growth is experimentally shown to be a function of the range of stress intensity ${\displaystyle \mathrm{\Delta }K}$ seen in a loading cycle. The Paris equation is^[1]

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where ${\displaystyle a}$ is the crack length and ${\displaystyle \mathrm{d}a/\mathrm{d}N}$ is the fatigue crack growth for a load cycle ${\displaystyle N}$. The material coefficients ${\displaystyle C}$ and ${\displaystyle m}$ are obtained experimentally and also depend on environmental effects, stress ratio, and the characteristic specimen size.^[2] The stress intensity factor range has been found to correlate the rate of crack growth from a variety of different conditions and is the difference between the maximum and minimum stress intensity factors in a load cycle and is defined as

${\displaystyle \mathrm{\Delta }K=K_{\text{max}}-K_{\text{min}}.}$

Being a power law relationship between the crack growth rate during cyclic loading and the range of the stress intensity factor, the Paris law can be visualized as a linear graph on a log-log plot, where the x-axis is denoted by the range of the stress intensity factor and the y-axis is denoted by the crack growth rate (see Figure 1).

The equation gives the growth for a single cycle. Single cycles can be readily counted for constant amplitude loading. Additional cycle identification techniques such as rainflow-counting algorithm need to be used to extract the equivalent constant amplitude cycles from a variable amplitude loading sequence.

• 2Domain of applicability

History[edit]

In a 1961 paper, P.C. Paris introduced the idea that the rate of crack growth may depend on the stress intensity factor.^[3] Then in their 1963 paper, Paris and Erdogan indirectly suggested the Paris law with the aside remark 'The authors are hesitant but cannot resist the temptation to draw the straight line slope 1/4 through the data..' after reviewing data on a log-log plot of crack growth versus stress intensity range.^[4] The Paris equation was then presented with the fixed exponent of 4.

Domain of applicability[edit]

Stress Ratio[edit]

Higher mean stress is known to increase the rate of crack growth and is known as the mean stress effect. The mean stress of a cycle is expressed in terms of the stress ratio${\displaystyle R}$ which is defined as

${\displaystyle R=\frac{K_{\text{min}}}{K_{\text{max}}},}$

or ratio of minimum to maximum stress intensity factors. In the linear elastic fracture regime, ${\displaystyle R}$ is also equivalent to the load ratio

${\displaystyle R\equiv \frac{P_{\text{min}}}{P_{\text{max}}}.}$

Fatigue Crack Growth Testing

The Paris-Erdogan equation does not explicitly include the effect of stress ratio, although equation coefficients can be chosen for a specific stress ratio. Other crack growth equations such as the Forman Equation do explicitly include the effect of stress ratio as does the Elber equation by modelling the effect using crack closure.

Intermediate stress intensity range[edit]

Paris' law holds over the mid-range of growth rate regime as shown in the figure 1, but does not apply for very low values of ${\displaystyle \mathrm{\Delta }K}$approaching the threshold value ${\displaystyle \mathrm{\Delta }{K}_{\text{th}}}$, or for very high values approaching the material's fracture toughness, ${\displaystyle K_{\text{Ic}}}$. The alternating stress intensity at the critical limit is given by as shown in the figure 1.^[5]

The slope of the crack growth rate curve on log-log scale denotes the value of the exponent ${\displaystyle m}$ and is typically found to lie between the range ${\displaystyle 2}$ to ${\displaystyle 4}$. For materials with low static fracture toughness such as high strength steels, the value of ${\displaystyle m}$ can be as high as ${\displaystyle 10}$.

Long cracks[edit]

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Because the size of the plastic zone ${\displaystyle (r_{\text{p}}\approx K_I^2/{\sigma }_y^2)}$ is low in comparison to the crack length, ${\displaystyle a}$ (here, ${\displaystyle {\sigma }_y}$ is yield stress), small scale yielding applies, enabling the use of linear elastic fracture mechanics and the stress intensity factor. Thus, Paris' law is also only valid in the linear elastic fracture regime, under tensile loading and for long cracks.^[6]

References[edit]

1. ^'The Paris law'. Fatigue crack growth theory. University of Plymouth. Retrieved 28 January 2018.
2. ^Roylance, David (1 May 2001). 'Fatigue'(PDF). Department of Materials Science and Engineering, Massachusetts Institute of Technology. Retrieved 23 July 2010.
3. ^P.C. Paris, M.P. Gomez, and W.E. Anderson. A rational analytic theory of fatigue. The Trend in Engineering, 1961, 13: p. 9-14.
4. ^P.C. Paris and F. Erdogan. A critical analysis of crack propagation laws. Journal of Basic Engineering, 1963.
5. ^Ritchie, R.O; Knott, J.F (May 1973). 'Mechanisms of fatigue crack growth in low alloy steel'. Acta Metallurgica. 21 (5): 639–648. doi:10.1016/0001-6160(73)90073-4. ISSN0001-6160.
6. ^Ekberg, Anders. 'Fatigue Crack Propagation'(PDF). Retrieved 6 July 2019.

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